CN-TST 2018 Problem 1

Let $p,q$ be positive reals with sum 1. Show that for any $n$ -tuple of reals $(y_1,y_2,...,y_n)$ , there exists an $n$ -tuple of reals $(x_1,x_2,...,x_n)$ satisfying $$p\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i$$ for all $i=1,2,...,2017$ , where $x_{2018}=x_1$ .